The Korteweg-de Vries (KdV) equation is a PDE which models a number of phenomena, including the propagation of waves in shallow water. Interest in the KdV equation blossomed in the 1960s with the discovery that it could be reformulated as a Lax equation involving the continuum Schroedinger operator. This led to the construction of an infinite family of KdV-like PDEs, known as the KdV hierarchy. In joint work with J. Christiansen, B. Eichinger, and P. Yuditskii, we show the global existence and almost-periodicity of classical solutions to the Cauchy problem for each KdV hierarchy for almost-periodic initial conditions under simple summability criteria on the spectral gap lengths of the associated Schroedinger operators.

In the first part of the talk, we consider a method which is called the polynomial inverse image method. It is used to transform results from an interval to more general subsets of the real line. This method has been successful in many situations: Bernstein's inequality, Markov's inequality, fine zero spacing of orthogonal polynomials, Widom factors for equilibrium measures, etc. In the second part, we focus on construction of Cantor sets obtained via composition of infinitely many polynomials. These sets are natural generalizations of classical polynomial Julia sets. We discuss recent results related to asymptotics of orthogonal polynomials and Chebyshev polynomials on them.

Dirichlet's Theorem on arithmetic progressions says that sequences of numbers like 2, 5, 8, 11, 14, 17, 20, etc contain infinitely many prime numbers (in this case 2, 5, 11, 17, etc). More precisely, it says that if an arithmetic progression has a first term and a common difference that share no prime factors, then within the progression there are infinitely many prime numbers. I will prove Dirichlet's theorem for the sequence 1, 5, 9, 13, 17, 21, etc, following the argument of Dirichlet's proof. The full proof involves a heavy mixture of complex analysis, group character theory, and tricky estimates. The particular case I will show contains all the features of the proof, but the details are simple enough to be understood by a student who has taken Math 102.

4:00 pm Monday, September 18, 2017Topology Seminar: The period mapping on outer space
by Neil Fullarton (Rice) in HBH 227

(This is joint work with Corey Bregman.) Given a basis of 1-cycles in a finite graph, we can construct an inner product on the graph's homology. Using this, we define the 'period mapping' from the moduli space of marked, metric graphs (Culler--Vogtmann's 'outer space') to the moduli space of marked, flat tori, two very well-studied spaces. I'll discuss basic properties of this map, and compare with the more classical period mapping for Riemann surfaces. Our main result explicitly determines the homotopy type of the fibers of this graph-theoretic period mapping. I'll also discuss how the space of marked, so-called 'hyperelliptic' graphs behaves as a branch locus for the mapping, and how we show that this locus is, in a sense, simply-connected at infinity.

4:00 pm Tuesday, September 19, 2017AGNT: Unnormalized differences of the zeros of the derivative of the completed L-function
by Arindam Roy (Rice University) in HBH 227

We study the distribution of unnormalized diferences between imaginary parts of the zeros of the derivative of the Riemann ξ function. Such distributions are capable of identifying the exact location of every zero of the Riemann zeta function. In particular, we prove that the Riemann hypothesis for the Riemann zeta function is encoded in the distribution of zeros of derivative of completed Dirichlet L-function. We also show that the differences tend to avoid the imaginary part of low lying zeros of the Riemann zeta function.

We derive a bound for the eigenvalue counting function (for strictly positive eigenvalues) for higher-order Krein Laplacians. The latter are particular self-adjoint extensions of minimally defined, positive integer powers of the Laplacian on arbitrary open, bounded sets. The bound extends to open, finite volume domains of finite width, subject to a compact Sobolev embedding property, and shows the correct high-energy power law behavior familiar from Weyl asymptotics. This is based on joint work with M. Ashbaugh, F. Gesztesy, A. Laptev, and M. Mitrea.

In this talk we will begin with a brief history of the mathematics of aperiodic tilings of Euclidean space, highlighting their relevance to the theory of physical materials called quasicrystals. Next we will focus on an important collection of point sets, cut and project sets, which provide us with mathematical models for quasicrystals. Cut and project sets have a dynamical description, in terms of return times to certain regions of linear R^d actions on higher dimensional tori. As an example of the utility of this point of view, we will demonstrate how it can be used, in conjunction with input from Diophantine approximation, to classify a subset of `perfectly ordered’ quasicrystals.

Every finite presentation P of a group G can be represented by two sets X,Y of pairwise disjoint, oriented simple closed curves on a closed oriented surface, S: generators correspond to components of X and relators to components of Y – which are written as words in the generators by their intersection patterns. This data also determines a compact pseudo (possibly real) 3-manifold M∗ and we observe that G is isomorphic to π_1(M^∗) ∗ F_{β_0(S−X)−1}. In particular every finitely presented group is the fundamental group of some pseudo 3-manifold. The free factor F_{β_0(S−X)−1} can be easily delt with and we see that the minimal genus among all such surfaces S gives an invariant g(P) which measures ”how far” M^∗ is from a 3-manifold. For G freely indecomposable M^∗ is a 3-manifold if and only if g(P) = β_0(X). Cognizant of the fact that simply defined invariants may turn out to be pretty much useless, I decided to see if I could compute any interesting examples before I publicized these ideas. I have done this for (standard presentations) of finitely generated Abelian groups, and will discuss these results.

4:00 pm Tuesday, September 26, 2017AGNT: On the period maps for certain Horikawa surfaces and for cubic pairs
by Zheng Zhang (Texas A&M)

It is an interesting problem to attach moduli meanings to locally symmetric domains via period maps. Besides the classical cases like polarized abelian varieties and lattice polarized K3 surfaces, such examples include quartic curves (by Kondo), cubic surfaces and cubic threefolds (by Allcock, Carlson and Toledo), and some Calabi-Yau varieties (by Borcea, Voisin, and van Geemen). In the talk we will discuss two examples along these lines: (1) certain surfaces of general type with p_g=2 and K^2=1; (2) pairs consisting of a cubic threefold and a hyperplane section. This is joint work with R. Laza and G. Pearlstein.

Given Riemannian manifolds $M$ and $N$, consider maps $f: M \rightarrow N$ and $g: M \rightarrow N$ which are homotopic and $L$-Lipschitz. Gromov asked the following question: Does there exist a homotopy from $f$ to $g$ which is itself $L$-Lipschitz? In this talk, I will describe recent work with D. Dotterrer, F. Manin, and S. Weinberger which partially answers this question. I will also outline some interesting applications of our results.

There are many interesting complex dynamical systems in physics, biology and other sciences. These systems have large sizes, and therefore very hard to solve. Continuum mechanics aims to reduce the size of the systems while still capturing the important macroscopic behaviors. In this talk, I will explain how mathematical models in continuum mechanics can be constructed to simplify the complex structures of large dynamical systems. We will explore four applications: (1) steering a cup of coffee; (2) traffic flows on highway; (3) tracking a hurricane; (4) modeling animal swarms.

Whyte introduced translation-like actions of groups which serve as a geometric generalization of subgroup containment. He then proved a geometric reformulation of the von Neumann conjecture by demonstrating a finitely generated group is nonamenable if and only if it admits a translation-like action by a non-abelian free group. This provides motivation for the study of what groups can act translation-like on other groups. As a consequence of Gromov's polynomial growth theorem, only nilpotent groups can act translation-like on other nilpotent groups. In joint work with David Cohen, we demonstrate if two nilpotent groups have the same growth, but non-isomorphic Carnot completions, then they can't act translation-like on each other.

4:00 pm Tuesday, October 3, 2017AGNT: Birational Geometry of moduli spaces of sheaves on surfaces and the Brill-Noether Problem
by Izzet Coskun (UIC) in HBH 227

In this talk, I will relate sharp Bogomolov inequalities on a surface to the ample cone of the moduli space of sheaves via Bridgeland stability. I will discuss weak Brill-Noether theorems on rational surfaces. As a consequence, I will describe the classification of Chern characters on the plane and on Hirzebruch surfaces such that the general bundle of the moduli space is globally generated. This is joint work with Jack Huizenga.

Lie groups provide a fertile source of Hamiltonian systems, both classical and quantum. The classical systems come via moment maps from the invariant polynomials of a Lie algebra, while the quantum systems come from the Harish-Chandra center of the enveloping algebra - for example, acting as commuting differential operators on locally symmetric spaces. I will explain how ideas of Kostant and Ngô (from the proof of the Fundamental Lemma) allow one to integrate the flows of all the resulting classical Hamiltonian systems. I will then show how this construction may be quantized, resulting in a new integration of quantum Hamiltonian systems. Time permitting I'll discuss our motivation, an application to the topology of character varieties of surfaces. (Based on joint work with Sam Gunningham.)

We investigate the maximal rate at which entanglement can be generated in quantum systems. The goal is to upper bound this rate. I will quickly review the problem in closed systems, and provide a simple proof of one of the upper bounds. In an open system the generator of irreversible dynamics consists of a Hamiltonian and dissipative terms in Lindblad form. The relative entropy of entanglement and quantum mutual information are chosen as a measure of entanglement in an open system. At the end I will discuss the most recent progress on a generalization of the entanglement rate problem, which, for example, provides the bound on entanglement rate for Renyi entropy.

The classical Riemann-Hilbert correspondence takes an ordinary differential equation in one complex variable to its corresponding monodromy representation. It is a classical problem to describe, as concretely as we can, what this correspondence actually is. In the last few years it has been understood that this problem is intimately connected with a web of other topics in geometry and quantum field theory. One key new ingredient is the theory of “generalized Donaldson-Thomas invariants”; originally introduced by Kontsevich-Soibelman and Joyce-Song in the study of 3-dimensional Calabi-Yau manifolds, these objects have turned out to be the key to a new scheme for solving the Riemann-Hilbert problem much more explicitly than was previously possible. I will review these developments, and some recent applications.

The $n$-solvable filtrations of the knot/link concordance groups were defined as a way of studying the structure of the groups and in particular, the subgroup of algebraically slice knots/links. While the knot concordance group $C^1$ is known to be an abelian group, when $m$ is at least $2$, the link concordance group $C^m$ of $m$-component (string) links is known to be non-abelian. In particular, it is well known that the pure braid group with m strings is a subgroup of $C^m$ and hence when $m$ is at least $3$, this shows that $C^m$ contains a non-abelian free subgroup. We study the relationship between the derived subgroups of the the pure braid group, $n$-solvable filtration of $C^m$, links bounding symmetric Whitney towers, and links bounding gropes. This is joint with with Jung Hwan Park and Arunima Ray.

Many finiteness and enumerative problems in number theory rely on the finiteness/enumeration of the set of solutions to the equation x+y=1 over the group of S-units in a number field, where Sis a finite set of primes. In 1995, Nigel Smart solved certain S-unit equations to enumerate all genus 2 curves defined over the rationals with good reduction away from p=2. Smart's work build on that of of Baker, de Weger, Evertse, Yu, and many others. In 2016, following Smart's methods, Malmskog and Rasmussen found all Picard curves over Q with good reduction away from p=3, and Angelos Koutsianas described methods for enumerating, and in some cases explicitly describes, all elliptic curves defined over a number field with good reduction outside S. Both projects required Sage implementation of special cases of Smart's general method. In January 2017, Alejandra Alvarado, Angelos Koutsianas, Beth Malmskog, Christopher Rasmussen, Christelle Vincent, and Mckenzie West combined these implementations and created new functions to solve the equation x+y=1 over the S-units of a general number field K for any finite set S of primes in K. The code is available on SageTrac and is under review for inclusion in future releases of Sage. This talk will give an overview of motivating problems and applications, the methods involved, and next steps to advance the theory and/or to improve this implementation.

We will define graph Laplacian in the most general form and address its basic properties, such as boundedness and essential selfadjointness. We will also investigate heat equation semigroup of such operators and it's stochastic completeness. The talk is based on the survey paper of Keller and Lenz "Unbounded Laplacians On Graphs: Basic Spectral Properties And The Heat Equation".

We prove that the class of subgroups of Diff^r(S^1) is not closed under taking free products for each 2<= r <=infinity. More specifically, if G is finitely generated non-virtually-abelian group, then (G x Z) * Z does not embed into Diff^r(S^1). We then complete the classification of RAAGs embeddable in Diff^r(S^1), answering a question of Kharlamov (in a paper of M. Kapovich). This is a joint work with Thomas Koberda.

The non-cuspidal points of X_1(N) correspond to isomorphism classes of pairs (E,P), where E is an elliptic curve and P is a point on E of order N. If E has complex multiplication by an order in an imaginary quadratic field K, we say (E,P) is a K-CM point. In this talk, I will give a compete classification of the degrees of K-CM points on X_1(N)_{/K}, where K is any imaginary quadratic field. This is joint work with Pete L. Clark.

Over the last 25 years we have seen incredible advances in the performance of end-user language technologies such as speech recognition and machine translation. However, almost all of the research and engineering effort to date has been expended on 100 or so languages, primarily those of greatest commercial interest: English, French, Chinese, Japanese, German, etc. We'll begin by explaining some of the mathematical models underlying this work, focusing on language modeling as a relatively simple but foundational technology. We'll move on to discuss some of the ways existing models fall short for non-Indo-European languages, the difficulties faced by small language groups in terms of resource-building, and our efforts to overcome some of these difficulties for the next thousand (or more) languages.

Let p and q are relatively prime positive integers with p less than q. A Ford circle C(p/q) is a circle lying in the upper half plane tangent to the point p/q on the real line with radius 1/2q^2. We will show you some interesting features of Ford circles. They never intersect each other. The sum of the area of Ford circles is computable and equal to (pi)(zeta(3))/4zeta(4). If time permits then we will describe the method to count the Ford circles up to a given radius.

Bowditch described the boundary of a relatively hyperbolic group pair $(G,P)$ as the boundary of any hyperbolic space that $G$ acts geometrically finitely upon, where the maximal parabolic subgroups are conjugates of the subgroups in $P$. For example, the fundamental group of a hyperbolic knot complement acts geometrically finitely on $\mathbb{H}^3$, where the maximal parabolic subgroups are the conjugates of $\mathbb{Z} \oplus \mathbb{Z}$. Here the Bowditch boundary is $S^2$. We show that torsion-free relatively hyperbolic groups whose Bowditch boundaries are $S^2$ are relative $PD(3)$ groups. This is joint work with Bena Tshishiku. If time permits, I'll show some examples of strange phenomena that can happen with boundaries of relatively hyperbolic groups, joint with Chris Hruska.

In 1977, Mazur showed that the "Eisenstein quotient" of J_0(p) has rank 0 (and so finitely many rational points). We show that for many primes p, there is a further quotient of J_0(p) such that a positive proportion of quadratic twists also have rank 0. This is a special case of a general result concerning abelian varieties with real multiplication. The proof uses recent our work with Manjul Bhargava, Zev Klagbsrun, and Robert Lemke Oliver, on the average size of the Selmer group of a 3-isogeny in any quadratic twist family.

Algebraic K-theory is a fundamental invariant encoding information about number theory, manifold geometry, and algebraic geometry. However, it is hard to compute directly. Instead, a very successful approach to studying algebraic K-theory has been via "trace methods", which map out to more tractable theories such as (topological) Hochschild and cyclic homology. These theories are interesting in their own right. In this talk, I will give a gentle overview of this story.

I will discuss the geometry and topology of complex hyperbolic 2-manifolds, highlighting open questions and recent progress directly inspired by the last 40 years of work on hyperbolic 2- and 3-manifolds. Emphasis will be on explicit topological constructions (particularly of minimal volume manifolds), fibrations, and betti numbers. Much of this will cover joint work with Luca Di Cerbo.

Arithmetic topology has its origins in an analogy between rings of integers and three dimensional manifolds which goes back to Barry Mazur in the 1960's. In this talk I will discuss some recent developments in the subject. One has to do with an arithmetic version of Chern Simons theory suggested by Minhyong Kim. Another has to do with the application of Massey triple products to Iwasawa theory. These analogies are a two way street. Results in number theory or about three manifolds suggest new questions about the other of the two subjects.

4:00 pm Tuesday, November 7, 2017AGNT: Reduction of dynatomic curves: The good, the bad, and the irreducible
by Andrew Obus (University of Virginia) in HBH 227

The dynatomic modular curves parameterize one-parameter families of dynamical systems on P^1 along with periodic points (or orbits). These are analogous to the standard modular curves parameterizing elliptic curves with torsion points (or subgroups). For the family x^2 + c of quadratic dynamical systems, the corresponding modular curves are smooth in characteristic zero. We give several results about when these curves have good/bad reduction to characteristic p, as well as when the reduction is irreducible. We will also explain some motivation from the uniform boundedness conjecture in arithmetic dynamics.

In this talk I will discuss continuity of eigenvalues and eigenfunctions of self-adjoint Schr\”odinger operators on metric graphs with respect to edge lengths. The standard results in this direction address only the case of strictly positive edge lengths. I will show that most of these results can be carried over to the case of zero limiting lengths.

4:00 pm Wednesday, November 8, 2017Colloquium: Complexity of triviality in topology with applications to geometric variational problems
by Alexander Nabutovsky (University of Toronto) in HBH 227

The first theme of the talk is that some geometric objects have trivial topology, but it is very difficult to see that this, indeed, is the case. The second theme is that this phenomenon implies the ruggedness of some moduli spaces in differential and combinatorial geometry, and that it even forces the existence of non-trivial solutions of some problems in geometric calculus of variations. These phenomena were previously known for dimensions >4 (joint work with Shmuel Weinberger). However, recently we managed to prove that they already exist in dimension 4 (joint work with Boris Lishak). In particular, I will show that (1) it is very difficult to untie some trivial 2-knots in the four-dimensional space; (2) there exist ``many" contractible 2-dimensional complexes, which are extremely difficult to contract and for each pair of them it is also very difficult to see that they are homotopy equivalent to each other; (3) for each n>3 there exist infinitely many distinct local minima of curvature-pinching sup |K| diam^2 (C^0 norm of the sectional curvature normalized by the square of the diameter) on the space of Riemannian structures on the n-sphere; (4) in high-dimensions there exist many non-trivial ``thick" knots of codimension one (unlike the usual knot theory).

Thirty five years ago M. Gromov asked if it is true that the length of a shortest periodic geodesic on a closed Riemannian manifold does not exceed c(n)Vol^{1/n}, where Vol denotes the volume of the manifold, and c(n) is a constant that depends only on its dimension n. This question and a similar question with the diameter of the manifold instead of Vol^{1/n} are still open. I will discuss the solutions to these and related questions in dimension 2, as well as the upper bounds for periodic geodesics, stationary geodesic nets, loops and minimal surfaces in higher dimensions.

4:00 pm Friday, November 10, 2017Undergraduate Colloquium: What I Did Last Summer
by Ilya Marchenka '19 and Anh Tran '19 (Rice University) in HBH 227

Come hear two Rice math majors tell you about their research experiences last summer! They will tell you about their projects and what they enjoyed about the experience. There will also be lots of time for questions so that you can learn about the opportunities available for you! Ilya implemented a prototype of a multiplicatively homomorphic, pairing-based cryptosystem to be used in elections. Anh studied pattern-avoiding permutations using algebraic tools such as characteristic polynomials of linear recurrences.

The tautological ring of the moduli space of curves is a subring of the Chow ring that, on the one hand, contains many of the classes represented by "geometrically defined" cycles (i.e. loci of curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By this I mean that we can explicitly describe a set of additive generators, which are indexed by suitably decorated graphs. The study of the tautological ring was initiated by Mumford in the '80s and has been intensely studied by several groups of people. Just a couple years ago, Pandharipande reiterated that we are making progress in a much needed development of a "calculus on the tautological ring", i.e. a way to effectively compute and compare expressions in the tautological ring. An example of such a "calculus" consists in describing formulas for geometrically described classes (e.g. the hyperelliptic locus) via meaningful formulas in terms of the combinatorial generators of the tautological ring. In this talk I will explain in what sense "graph formulas" give a good example of what the adjective "meaningful" meant in the previous sentence, and present a few examples of graph formulas. The original work presented is in collaboration with Nicola Tarasca and Vance Blankers.

In this talk I will discuss continuity of eigenvalues and eigenfunctions of self-adjoint Schr\”odinger operators on metric graphs with respect to edge lengths. The standard results in this direction address only the case of strictly positive edge lengths. I will show that most of these results can be carried over to the case of zero limiting lengths.

I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of R^4 to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with András Vasy.

4:00 pm Wednesday, November 15, 2017Geometry-Analysis Seminar: Stability results of generalized Beltrami fields and applications to vortex structures in the Euler equations
by David Poyato (University of Granada) in HBH 227

Strong Beltrami fields, that is, 3D velocity fields whose vorticity is the product of itself by a constant factor, are particular solutions to the Euler equations that have long played a key role in Fluid Mechanics. Its importance relies on the Lagrangian theory of turbulence as they were expected to exhibit chaotic configurations. Very recently, this ancient conjecture of Lord Kelvin was positively answered by Alberto Enciso and Daniel Peralta-Salas (ICMAT, Spain). Specifically, there are strong Beltrami fields exhibiting any type of linked vortex lines and tubes of arbitrarily complicated topology. Nevertheless, such existence result is quite tight in the sense that Beltrami fields with non-constant factor (generalized Beltrami fields) are “rare”. Thus, the existence of turbulent configurations is limited to Beltrami fields with a constant factor. The aim of this talk is twofold. First, we will review the state of the art in this topic. Second, we will show that although a full stability result is not possible, there are certain privileged ways to perturb a strong Beltrami field and obtain Beltrami fields with a non-constant factor that even realize arbitrarily complicated vortex structures. This partial stability will be captured in terms of an "almost global” and a “local" stability theorem. The proof relies on analyzing the well-posedness and propagation of compactness and regularity of an innovative iterative scheme of Grad-Rubin type inspired by some numerical methods coming from Astrophysics.

The Brauer dimension of a field F is defined to be the least number n such that index(A) divides period(A)^n for every central simple algebra A defined over any finite extension of F. One can analogously define the Brauer-p-dimension of F for p, a prime, by restricting to algebras with period, a power of p. The 'period-index' questions revolve around bounding the Brauer (p) dimensions of arbitrary fields. In this talk, we look at the period-index question over complete discretely valued fields in the so-called 'bad characteristic' case. More specifically, let K be a complete discretely valued field of characteristic 0 with residue field k of characteristic p > 0 and p-rank n (= [k:k^p]). It was shown by Parimala and Suresh that the Brauer p-dimension of K lies between n/2 and 2n. We will investigate the Brauer p-dimension of K when n is small and find better bounds. For a general n, we will also construct a family of examples to show that the optimal upper bound for the Brauer-p-dimension of such fields cannot be less than n+1. These examples embolden us to conjecture that the Brauer p-dimension of K lies between n and n+1. The proof involves working with Kato's filtrations and bounding the symbol length of the second Milnor K group modulo p in a concrete manner, which further relies on the machinery of differentials in characteristic p as developed by Cartier. This is joint work with Bastian Haase.